On the lattice of principal generalized topologies (Q2416503)

In characterizing principal generalized topologies, the concept of ultratopology is used in the paper. For an ultratopology we require such an ultrafilter $\mathcal{U}$ on $X$ that $x \in X$ is not in $\bigcap \mathcal{U}$; the topology $\tau(x, \mathcal{U})=\{G \mid x\in G^{c}\, {\text or}\,\, G \in \mathcal{U}\}$ is called an ultratopology; if $\mathcal{U}$ is a principal ultrafilter then $\tau(x, \mathcal{U})$ is said to be a principal ultratopology on $X$. It is proved that a generalized topology $\mu$ on $X$ is $T_{1}$ iff it is the meet of non-principal ultratopologies in $\mathcal{G}T(X)$ ($\mu$ and $\mathcal{G}T(X)$ are standard notations).\par A generalized topology $\mu$ on $X$ is called principal (PGT, in short) if $\mu=\mathcal{P}(X)$ or $\mu=\wedge\{\tau(x, \mathcal{U}(y))\mid \mu\leq \tau(x, \mathcal{U}(y))\}$ for $y\neq x$, where $\tau(x, \mathcal{U}(y))$ are the ultratopologies in $\mathcal{G}T(X)$. It is shown that for $\mu \in P\mathcal{G}T(X)$, and for each $x\in X$, the set $B_{x}=\{y \in X \mid y\leq \tau(x, \mathcal{U}(y))\}$ is $\mu$-open; it is further shown that every $\mu$-open set containing $x$ must also contain the set $B_{x}$. Finally, for $P\mathcal{G}T(X)$, it is shown that such a collection is a meet-complete, bounded sublattice of the lattice $\mathcal{G}T(X)$; however, $P\mathcal{G}T(X)$ is not a complete sublattice of the lattice $\mathcal{G}T(X)$. For $S\subset X$, where $X$ is a set with a preorder relation $R$ on $X$, $S$ is called a component of $X$ relative to $R$ if for every pair $x, y \in S$ there exists a path from $x$ to $y$ in $R$ (suitably defined) and for $z \in (X \setminus S)$ there is no path from $t$ to $z$ for all $t \in S$. A preorder relation $R$ on $X$ is connected if $X$ has only one component relative to $R$. It is proved that a principal generalized topology $\tau_{R}$ on $X$ is connected iff $R$ is a connected preorder relation; further, it is proved that the lattice $\mathcal{G}T(X)$ is anti-isomorphic to the lattice $\mathcal{R}$ of preorder relations on $X$.\par Finally, in dealing with the concept of complementation in the lattice of principal generalized topologies, it is proved that the lattice $\mathcal{R}$ of preorder relations on $X$ is a complemented lattice, the lattice $P\mathcal{G}T(X)$ of principal generalized topologies on $X$ is a complemented lattice, and every generalized topology on $X$ has a principal complement.